{"id":"https://openalex.org/W2999073519","doi":"https://doi.org/10.1109/tsp.2020.3014611","title":"Deep Neural Networks With Trainable Activations and Controlled Lipschitz Constant","display_name":"Deep Neural Networks With Trainable Activations and Controlled Lipschitz Constant","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W2999073519","doi":"https://doi.org/10.1109/tsp.2020.3014611","mag":"2999073519"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2020.3014611","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2020.3014611","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2001.06263","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031317746","display_name":"Shayan Aziznejad","orcid":"https://orcid.org/0000-0002-6871-9703"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Shayan Aziznejad","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomedical Imaging Group, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-6871-9703","affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomedical Imaging Group, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008465855","display_name":"Harshit Gupta","orcid":"https://orcid.org/0000-0002-2910-9072"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Harshit Gupta","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomedical Imaging Group, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-2910-9072","affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomedical Imaging Group, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047627056","display_name":"Joaquim Campos","orcid":"https://orcid.org/0000-0002-6723-0295"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Joaquim Campos","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomedical Imaging Group, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomedical Imaging Group, Ecole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5027830035","display_name":"Micha\u00ebl Unser","orcid":"https://orcid.org/0000-0003-1248-2513"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Michael Unser","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne,"],"raw_orcid":"https://orcid.org/0000-0003-1248-2513","affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole polytechnique f\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne,","institution_ids":["https://openalex.org/I5124864"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I5124864"],"apc_list":null,"apc_paid":null,"fwci":0.4758,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.52454616,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"68","issue":null,"first_page":"4688","last_page":"4699"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9923999905586243,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.9350170493125916},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.7112193703651428},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6798185706138611},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6483267545700073},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.635418176651001},{"id":"https://openalex.org/keywords/piecewise-linear-function","display_name":"Piecewise linear function","score":0.5880911946296692},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.4942786693572998},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4771249294281006},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.47697004675865173},{"id":"https://openalex.org/keywords/spline","display_name":"Spline (mechanical)","score":0.4400220513343811},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4251546263694763},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4113270044326782},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1835743486881256},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17827630043029785},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06831705570220947}],"concepts":[{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.9350170493125916},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.7112193703651428},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6798185706138611},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6483267545700073},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.635418176651001},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.5880911946296692},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.4942786693572998},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4771249294281006},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.47697004675865173},{"id":"https://openalex.org/C10390562","wikidata":"https://www.wikidata.org/wiki/Q581809","display_name":"Spline (mechanical)","level":2,"score":0.4400220513343811},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4251546263694763},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4113270044326782},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1835743486881256},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17827630043029785},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06831705570220947},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1109/tsp.2020.3014611","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2020.3014611","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2001.06263","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.06263","pdf_url":"https://arxiv.org/pdf/2001.06263","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2999073519","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2001.06263","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:infoscience.epfl.ch:280139","is_oa":true,"landing_page_url":"https://infoscience.epfl.ch/handle/20.500.14299/171802","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"research article"},{"id":"doi:10.48550/arxiv.2001.06263","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2001.06263","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2001.06263","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.06263","pdf_url":"https://arxiv.org/pdf/2001.06263","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5733285100","display_name":null,"funder_award_id":"200020_184646/1","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G8773745792","display_name":null,"funder_award_id":"692726","funder_id":"https://openalex.org/F4320334678","funder_display_name":"European Research Council"}],"funders":[{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"},{"id":"https://openalex.org/F4320334678","display_name":"European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":62,"referenced_works":["https://openalex.org/W360270745","https://openalex.org/W606515514","https://openalex.org/W1498436455","https://openalex.org/W1533861849","https://openalex.org/W1540155273","https://openalex.org/W1594563152","https://openalex.org/W1677182931","https://openalex.org/W1683511521","https://openalex.org/W1901129140","https://openalex.org/W1986280275","https://openalex.org/W2003410902","https://openalex.org/W2015904350","https://openalex.org/W2050834445","https://openalex.org/W2055522016","https://openalex.org/W2103496339","https://openalex.org/W2112796928","https://openalex.org/W2143956139","https://openalex.org/W2144513243","https://openalex.org/W2144856549","https://openalex.org/W2146766088","https://openalex.org/W2148603752","https://openalex.org/W2149593872","https://openalex.org/W2151467704","https://openalex.org/W2153796359","https://openalex.org/W2156387975","https://openalex.org/W2161388792","https://openalex.org/W2162870748","https://openalex.org/W2163605009","https://openalex.org/W2243397390","https://openalex.org/W2286501826","https://openalex.org/W2290904946","https://openalex.org/W2338343974","https://openalex.org/W2555745781","https://openalex.org/W2574952845","https://openalex.org/W2595294663","https://openalex.org/W2618530766","https://openalex.org/W2751669091","https://openalex.org/W2767471303","https://openalex.org/W2911290743","https://openalex.org/W2919115771","https://openalex.org/W2939947188","https://openalex.org/W2963455631","https://openalex.org/W2963739978","https://openalex.org/W3038830718","https://openalex.org/W3103372211","https://openalex.org/W3103915668","https://openalex.org/W4298876635","https://openalex.org/W6629815555","https://openalex.org/W6631943919","https://openalex.org/W6637359451","https://openalex.org/W6639824700","https://openalex.org/W6681151457","https://openalex.org/W6681187884","https://openalex.org/W6682319494","https://openalex.org/W6682889407","https://openalex.org/W6683595889","https://openalex.org/W6695677705","https://openalex.org/W6729999211","https://openalex.org/W6735307612","https://openalex.org/W6740483536","https://openalex.org/W6748471204","https://openalex.org/W6779669310"],"related_works":["https://openalex.org/W3100643896","https://openalex.org/W2921395510","https://openalex.org/W3204582391","https://openalex.org/W2752253833","https://openalex.org/W2984810828","https://openalex.org/W2995824857","https://openalex.org/W2973114996","https://openalex.org/W2595751897","https://openalex.org/W3036251984","https://openalex.org/W20823384","https://openalex.org/W2947012451","https://openalex.org/W3198889502","https://openalex.org/W2111029549","https://openalex.org/W2953115901","https://openalex.org/W3012479795","https://openalex.org/W2943540325","https://openalex.org/W2914902801","https://openalex.org/W2798989470","https://openalex.org/W1535888473","https://openalex.org/W3202491425"],"abstract_inverted_index":{"We":[0,122],"introduce":[1],"a":[2,43,61,85,100],"variational":[3,62,69],"framework":[4],"to":[5,17,99],"learn":[6],"the":[7,19,22,29,34,47,55,96,111,114,117,141],"activation":[8,66],"functions":[9],"of":[10,21,28,33,50,113,119,144],"deep":[11],"neural":[12,51],"networks.":[13,52],"Our":[14,68],"aim":[15],"is":[16,71,74],"increase":[18],"capacity":[20],"network":[23,130],"while":[24],"controlling":[25],"an":[26,104],"upper-bound":[27],"actual":[30],"Lipschitz":[31,48],"constant":[32,49],"input-output":[35],"relation.":[36],"To":[37],"that":[38,81,87],"end,":[39],"we":[40,58,79,138],"first":[41],"establish":[42],"global":[44],"bound":[45],"for":[46,64],"Based":[53],"on":[54,110],"obtained":[56],"bound,":[57],"then":[59],"formulate":[60],"problem":[63,70,98],"learning":[65,118],"functions.":[67],"infinite-dimensional":[72],"and":[73,90,131,135,137],"not":[75],"computationally":[76],"tractable.":[77],"However,":[78],"prove":[80],"there":[82],"always":[83],"exists":[84],"solution":[86],"has":[88],"continuous":[89],"piecewise-linear":[91],"(linear-spline)":[92],"activations.":[93],"This":[94],"reduces":[95],"original":[97],"finite-dimensional":[101],"minimization":[102],"where":[103],"\u2113":[105],"<sub":[106],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[107],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">1</sub>":[108],"penalty":[109],"parameters":[112],"activations":[115],"favors":[116],"sparse":[120],"nonlinearities.":[121],"numerically":[123],"compare":[124],"our":[125,145],"scheme":[126],"with":[127],"standard":[128],"ReLU":[129],"its":[132],"variations,":[133],"PReLU":[134],"LeakyReLU":[136],"empirically":[139],"demonstrate":[140],"practical":[142],"aspects":[143],"framework.":[146]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}
